《数学与人类文明》Mathematics & Civilizations


1. 哲学:希腊 Euclidean Geometry

2. 艺术 : Arabic 文艺复兴, 达芬奇, Golden Ratio

3. 工业革命:Descartes Analytic Geometry, Newton Calculus

4. 抽象: Gauss “Non-Euclidean Geometry” , Paradox in Set Theory, Godel “Incomplete Theorem” .




1. 秦汉:张苍&耿寿昌 编 《九章算术》, 3AD 东汉 刘徽 (注解)

2. 4AD 南北朝:祖冲之父子 圆周率π

3. 13-16世纪 元/明朝:珠算盘


只是Applied 应用, 没有 希腊Deductive Theory 理论。




[0. 希腊 = 达摩祖师]

  1. 法国 =少林派
  2. 德国 = 武当派(德国祖师 Lindemann 是法国宗师 Charles Hermite 的得意外国徒弟
  3. 俄国 = 峨眉派 (俄国祖师奶 Sofya Vasilyevna Kovalevskaya 是 德国大师 Karl Weierstrass 的得意女弟子)
  4. 中国 = 古墓派 (‘小龙女’ 的奇特功夫 自成一派:中国古代数学 + 融合 留 法/英/美/日 数学派 )


英/美/日 数学都是支派,非主流宗派 掌门:

英国 受法国影响 : Newton 说他是站在巨人 Descartes, Fermat 的肩膀. 近代 Bertrand Russell 也是读法国Camile Jordan 的 《Group Theory 》书。

美国 数学 : WW2 德国/东欧 犹太人移民: Noether, Godel, Courant, Artin,…

日本数学 : 唐朝留学 中国 (日本”和算” 鼻祖 关孝和 把中国 “行列式” 传给欧洲Leibniz : Determinant ) ,明治天皇 时期 日本留学欧洲。


1. 奇特数学: 东汉 3AD  Chinese Remainder Theorem (韩信点兵 Modular Arithmetics), Singapore Modelling Math (eg. 鸡兔问题) ,13CE 的 行列式 (Determinant, before “Matrix” was invented in 20th CE by JJ Sylvester )

2. Determinant was invented by the ancient Chinese Algebraists 李冶 / 朱世杰 /秦九韶 in 13th century (金 / 南宋 / 元) in《天元术》.The Japanese “和算” mathematician 关孝和 spread it further to Europe before the German mathematician Leibniz named it the “Determinant” in 18th century. The world, however, had to wait till the 19th century to discover the theory of Matrix 矩阵 by JJ Sylvester (Statistical Math private Tutor of Florence Nightingale, the world’s first nurse) closely linked to the application of Determinant.

Dr. Eugenia Cheng: “How to Bake Pi”

Key Points of the Talk:

1) Math is interesting but is only so after undergraduate school. Before that, Math is taught as computation subject from Elementary to High school.

2) Braid : Bach music, Juggling 3 balls

3) Platonic Icosahedral (20面体) Structure discovered by ancient Greek Plato 2000 years ago, but can’t find a real world Icosahedral object until in Viruses found by Louis Pasteur in 20th century – also now in Sars, Covid19.

4) Group Theory : Battenberg Cake, Bed Mattress Rotate/Flop/ Flipping

5) Mobius Strip & Donnut Cutting.

6) Fermat’s Last Theorem : Andrew Wiles in 1994 proved in 7 years still with a “hole”, but fixed a year later by himself & his student.

7) MacLane Pentagon : Higher-Dimension Categories (PhD Math)

The Black-Sholes Formula – 诺贝尔经济奖Scholes, Merton与LTCM S

Key Points:

1) From 1950 an unknown PhD Math Thesis by French Dr. Bachelier (a PhD student of the 20CE last Polymath Henri Poincaré, who was not impressed with the ‘gambling’ math, gave only an above-average marks to the thesis) – invention of “Options” Trading to eliminate risk in stock fluctuations.

2) The technique is called “Dynamic Hedging”…

3) …by applying the Japanese mathematician “Ito Math”.

4) Noble Prize Economics awarded to “ Black-Sholes Formula”

5). Sholes & Merton Company “LTCM” – making tons of money until… 1997 Asian Crisis, bailed out by USA government after loss of billions !


Chinese Remainder Theorem 《韩信点兵 》& Ideal


Chinese Remainder Theorem

By 1930, even Bourbaki founder Dieudoné didn’t know “Ideal” when he read 《Algebra》from Noether’s lecture-note compiled by her student Van der Waeden


The Mathematical Revolution That Was Bred on a Sheep Farm | OpenMind


In 17CE a similar plague like COVID19 forced a young man Newton to be quarantined in a sheep farm for 2 years, he changed the universe with Differential & Integration Calculus, to “move” the planets.

Will COVID19 give birth to another “Newton” in some “kampong” at one corner in USA or China or Europe ?


Cédric Vilani (Chinese) Interview

French Fields Medalist Cédric Vilani 中文interview: 他苦思证明数学/物理定理,在第 1001 th 夜 @4am,”好像上帝给他打电话 – Un coup de fil du Dieu” , 突然开窍…

他从政加入Macron的小党 “En March”,当MP, 今年竞选 Paris Mayor.

2017 年 他引入 “Singapore Math” 进法国小学。


NLB Library has 13 copies of Cedric Vilani’s book for public loan.


Key Points:

  1. Three Ancient Greek Problems : a) Trisect an angle“, b) Doubling the Cube, c) Square a Circle
  2. Field Theory
  3. Abel
  4. Galois
  5. Proved by Wantzel 2 out of 3 Greek Problems: 1a) & 1b)

Note :1837 Pierre Wantzel proved 2 out of the 3 ancient Greek Problems “Trisect an angle” & Double a Cube, only 5 yrs after Galois had published his “Group & Field” Theories before death at 21.

Wantzel was the only person in French history who scored 1st in both Concours (法国抄袭中国的数学“科举“) of Ecole Normale & Ecole Polytechnique.
His life regret was switching from his talent in Math to becoming a mediocre Civil Engineer.


数学的故事 The Story of Math

This series of Math Story is excellent for students who are scrared of Math.

Math should be taught like History, Literature, Music, Art… together with the stories of Mathematical evolution since ancient time, the life of the great mathematicians who discovered them … through this teaching would students relate the “cold” textbook Math with Math beauty.

Marcus du Sautoy

  • Oxford Math Prof
  • Popular Math Author of the Best Seller《Music of the Primes 》(Loan at National Library Branches – Math Shelf #510. xx) .

Egyptian Math : Divide 9 loaves of bread among 10 persons (ie 9/10)

\boxed{\frac{1} {2} +\frac{1} {3}+ \frac{1} {15} = \frac{15+10+2} {30} =\frac{27} {30}=\frac{9} {10}}

Babylonian Quadratic Equation by Squaring an Area:

\boxed{X^{2} + 6X - 55 = 0}

Note: This is especially true to the University Math “Abstract Algebra” aka “Modern Math” – yet it is an “old lady” evolved 200 years ago from France during the French Revolution by Evariste Galois “Group Theory”. Most French students are scared of Abstract Algebra (a killer subject in French engineering school) even though we were good in High school Math (A-level or Baccalaureate). It was only 28 years after my university graduation from Math “Classe Préparatoire” that I got rid of the fear of (Abstract) Math, thanks to another Best Seller Popular Math Book 《Unknown QuantityA real & imaginary history of Algebra》by Prof John Derbyshire [Loan at NLB #510.xx]



The Math of Higher Dimensionality – Imaginary Numbers: Gauss & Riemann

\boxed { \sqrt{-1} = i}

谁说数学家不时尚 ? Cédric Villani



Key Points:

  • Classe Preparatoire (预科班) – most rigorous and toughest math training (2 years) after high school.
  • Grande Ecole (大学校)
  • “Bourbaki school” – secretive Math club still exists today.
  • French Thinking
  • Ecole Normale Superieure where he spent 8 years (Graduate, PhD, Teaching assistant)
  • His PhD Advisor Pierre Louis Leon (Fields Medalist)
  • American universities use “Engineering Way” to train mathematicians, French is different.
  • IMO Math vs French Abstract Math
  • Paris is still a World Math Center

让数学史融入初中数学教学 Math History in Math Curriculum


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不擅长考试的大数学家—法国数学家埃尔米特 Charles Hermite

【不擅长考试的大数学家—法国数学家埃尔米特 Charles Hermite】

He was 15 years junior of Galois from the same Math teacher Richard of Lycee Louis Le Grand. Richard gave him Galois’s student math homework books.

Charles Hermite proved ‘e’ transcendental, his German student Lindermann followed his method proved “pi” also transcendental.

Lindermann produced students Felix Klein, who produced Gauss, Riemann, David Hilbert…

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小算盘 大乾坤 Abacus

浙江临海市 国华珠算博物管主人 木匠 雷国华 收藏算盘20年,凭个人力量,保存中国算盘发源地的文化遗产,为此散尽钱财借债,免费开放给公众参观。说到家人因支持他而受苦,感慨激动不已。

中国的算盘上面2珠 (2×5),下面5珠(5), 共15,加1 = 16 进位。古代是16位 (Hexadecimal) 制, 半斤 八两 (= 1/2 斤 x 16 两 = 8 两)。

日本改良成10位制: 上面1珠 (1×5),下面4珠(4), 共9,加1 = 10 进位。

明朝万历年间的 “平民王子”朱载堉 (1536 AD – 1610 AD) 发明现代音乐的 十二平均律 (12-tone Equal Temperament) , 用81档的大算盘算出:先开立方根,后开平方根 2 次

\boxed { \displaystyle \sqrt[12] {2} = 2^{\frac {1}{12} }= 2^{{\frac {1}{3}}.{\frac {1}{2}}.{\frac {1}{2}}} = \sqrt {\sqrt {\sqrt[3]{2}}}}

The most addictive theorem in Applied mathematics

What is your favorite theorem ?

I have 2 theorems which trigger my love of Math :

  1. Chinese Remainder Theorem: 韩信点兵, named after a 200 BCE Han dynasty genius general Han Xin (韩信) who applied this modern “Modular Arithmetic” in battle fields.
  2. Fermat’s Last Theorem:The Math “prank” initiated by the 17CE amateur Mathematician Pierre de Fermat kept the world busy for 380 years until 1974 proved by Andrew Wiles.

Note 1: Lycée Pierre de Fermat (Classe Préparatoire) happens to be my alma mater named after this great Mathematician born in the same southern France “Airbus City” Toulouse.

Note 2: His another Fermat’s Little Theorem is used in modern computer cryptography.


Fields Medals 2018

Picture Order (From Left) :1. 2. 3. 4.

1. Caucher Birkar (UK / Kurdish – Iran, 40)


2. Alessio Figalli (Italy, 34)

3. Akshay Venkatesh (Australia / India, 36)

Studies number theory and representation theory.


4. Peter Scholze (Germany, 30)
Intersection between number theory and geometry


这位中国留英数学专业的小伙子有很好的口才: 数学之美, 他的数学遗憾。

华威大学(The University of Warwick),famous for mathematics in UK.

Key Points:

Einstein用 Riemann Geometry 数学救了Newton 物理。

中国数学的没落: 自从明朝科举废除数学考试

3位大师救近代中国数学: 华罗庚, 陈省身, 苏步青

数学光明的未来: AI, Big Data, Cloud Computing

天才的崎岖道路: “扫地僧”张益唐

时代的”指数” exponential快脚步, 不要追风, 要一以贯之 : Prof. Andrew Wiles proved 350-year-old “Fermat’s Last Theorem” (FLT).

[纠正]: Andrew Wiles 超过40岁, 没赶上Fields Medal, 只得个”奖励”。他看到椭圆(Ellipse)气球, 得到突破 FLT “工具”的灵感 – “Elliptic Curve” 。

数学是什么: 爱情, 艺术, 音乐, 科技

Visionary mathematician Vladimir Voevodsky

A self-study Russian mathematician, kicked out 3 times in high schools, expelled from Moscow University, all because he did not attend classes, preferred to self-study in a broader scope for his curiosity, at his own faster speed than the rigid curriculum and boring test-and-exams regime in classrooms.

He did the PhD in Harvard by invitation even he did not have a Bachelor degree, and he barely passed the Harvard’s QE (Qualifying Exams) in Algebraic Geometry, a field in which he made a revolutionary discovery few years later, and for which he was awarded the highest honor : Fields Medal.

This is the typical trait of the geniuses like Evariste Galois, Albert Einstein, Ramanujian, Hua Luogeng (华罗庚), Zhang YiTang (张益唐 – proved “70m Twin Prime Gap”) [#] , Chen Jingrun (陈景润, proved Goldbach Conjecture “1+2”) [##]… with self-motivated curiosity in their field of passion, with reading from the Masters’ works by themselves, PLUS thinking on a problem over many years with perseverance, they finally made great discoveries !


  1. Topology 拓扑
  2. Algebraic Geometry 代数几何: Geometry and Algebraic Equations
  3. Motif: Grothendieck
  4. Homotopy Theory 同伦

[Full Article] :


[#] : https://tomcircle.wordpress.com/2014/03/01/gaps-between-primes-zhang-yitang/

[##] : https://tomcircle.wordpress.com/2015/07/24/goldbach-conjecture-theorem-chen/

How (not) to memorise mathematics

Many excellent Math students after leaving universities more than 10 years forget 90% of math they learned, save some primary school arithmetics – few could do Singapore PSLE Modelling Math or solve quadratic equations.

The “Story-Telling” memory technique via “Signposts” can be used to reconstruct math from first principles:


Note: Lewis Carroll: the author of “Alice in wonderland”

Cambridge Professor Tim Gowers (Fields Medalist) suggested the similar pedagogy of “Memorise by First Principles”.

What is math? 

What is Math ? Interesting article below:


  • Mathematics = “that which is learned“ –(Pythagoras)

Math is not about calculation, it is understanding the nature, the universe, the philosophy (logic, intelligence – both “human” and “artificial”)…

What is Axiom, Lemma, Proposition ? Why rigorous Calculus was needed hundred years after Newton & Leibniz had invented it – “Epsilon-Delta” Analysis.

Difference between Riemann Integral & Lebesgue Integral ?

Unknown German Retiree Proved The “Gaussian Correlation Inequality” Conjecture


\boxed {P (a + b) \geq P (a) \times P (b)}

Case “=” : if (a, b) independent 
Case “>” : if (a, b) dependent

Thomas Royen used only high-school math (function, derivative) in his proof in 2014. He then published it in arxiv.org website – like Perelman did with the “Poincaré Conjecture”.

The genius who rejected Fields Medal & Clay Prize: Grigori Perelman 

Russian Jewish mathematician Grigori Perelman proved the Poincaré Conjecture in 7 years of solitude research in his Russian apartment  – same 7 years of solitude for Andrew Wiles (The Fermat’s Last Theorem) in the Cambridge attic house and Zhang Yitang  张益唐 (7-Million-Gap Twin Primes) in the “Subway” sandwich kitchen.

7 is the Perfect Number. 1 week has 7 days,  according to the “Book of Genesis”, God created the universe in 6 days and rested “Sabbath” on the beautiful 7th day.

People involved in his journey: 

  • His mother who supported his early education in Math, all the way to his international fame and adulthood — Erdös Paul also had a mother supporting her single son in the entire Mathematical life.
  • Prof Hamilton who “unintentionally” gave him the inspiration of the “Ricci-flow” tool – the first key to the door of The Poincaré Conjecture.
  • His closed Chinese friend the MIT Math Prof  田刚 (Tian Gang) – a PhD student of Prof S.T. Yau.
  • Fields Medal which he rejected — reason being he felt Prof Hamilton deserved a share of the credit.
  • Clay Prize which he rejected the U$ 1 million — for the same reason as the Fields Medal.
  • Prof  S.T. Yau (邱成桐) ‘s lawsuit against “The New Yorker” jounalists on “The Poincaré Conjecture” article regarding the purported claims by his 2 Chinese mathematician students.
  • Trump’s Speaking Math Formula

    The lower in the score the better : Trump (4.1) beats Hillary (7.7) who beats Sanders (10.1)

    Trump defied most expectation from the world to win the 2017 President of the USA. His victory over the much highly educated Ivy-league Yale lawyer-trained Hilary Clinton who speaks sophisticated English is “SIMPLE English“: seldom more than 2-syllable words.

    1-syllable words mostly: eg.dead, die, point, harm,…

    2-syllable words to emphasize: eg. pro-blem, ser-vice, bed-lam (疯人院), root cause, …

    3-syllable words to repeat (seldom): eg. tre-men-dous

    His speech is of Grade-4 level, reaching out to most lower-class blue-collar workers who can resonate with him. That is a powerful political skill of reaching to the mass. Hilary Clinton’s strength of posh English is her ‘fatal’ weakness vis-a-vis connecting to the mass.

    In election time, it is common to see candidates who win the heart of voters by using the local dialects of the mass, never mind they are discouraged in schools or TV: Hokkien, Teochew, etc.

    Curious Thoughts in Math & Science 

    1. Statistical Mechanics: e^ {- Ht}

    Quantum Mechanics: e^{iHt}

    2. Ramanujian:

    1 +2 + 3 + …+ n = –  1/12

    Note: this formula is used in Quantum Physics dealing with infinity n (although it cancels out each other in subsequent calculations)

    Tau Special Function:

    \boxed {\displaystyle \sum_{n=1}^{\infty}\tau (n) x^{n} = x \{(1-x)(1-x^{2})(1-x^{3})... \}^{24}}

    3. Boolean Algebra: George Boole (1847 in 《The Mathematical Analysis of Logic》) used Symbolic variables (not numbers) for Logic, inspired by Galois (1832 in Groups & Finite Fields), Hamilton’s quaternion algebra (1843).

    AND\boxed {x.y} 

    NOT\boxed {1-x} 

    XOR\boxed {x+y-2x.y} 

    Extra constraints ”  \boxed {x^{2}=x} 

     4. Solomon Golomb, Sol: “Linear Feedback Shift Register” (LFSR)  — shift left the first register, fill in the back register with XOR of certain “Taps” (eg.chosen the 1st, 6th, 7th registers)

    Maximal Length = The shift register of size n will repeat every 2^{n}-1 steps (exclude all ‘0’ sequence).

    Which arrangement of “Taps” will produce the maximal length ? 

    Solomon applied Pure Math : represent the above sequence of  registers algebraically  by: 

    \boxed {x^{7}+x^{6}+ 1} 

    in reducible modulo 2 (prime in polynomial, ie can’t be factored).

    => the sequence is Maximal length

    LFSR Applications: Telecommunications, 3G/4G/5G, CDMA, Wifi, computers, network, signal transmission error-correction CD/DVD, Astrology : Venus-Earth distance,  etc.

    A Day in the Mathematician’s Life 

    “Mathematician is a person who turns coffee into Theorems” – Erdös Paul , the Hungarian Mathematician who had no home, no wife, travelled around the world to visit his math collaborators proving 1000+ theorems. 数学界的”丐帮洪七公”! 他是华罗庚的好友, 和 陈省身 同年得 数学Wolf Prize.